First, you have "below" and "above" reversed. The plane z= 3- 2y is above the paraboloid [itex]z= x^2+ y^2[/itex] in the bounded region.

They intersect where [itex]z= x^2+ y^2= 3- 3y[/itex] which is equivalent to [itex]x^2+ y^2+ 3y= 3[/itex]. [itex]x^2+ y^2+ 3y+ 9/4= x^2+ (y+3/2)^2= 3+ 9/4= 21/4[/itex]. That is a circle with center at (0, -3/2) which is why ordinary polar coordinates do not give a simple equation. Either
1: integrate with x from [itex]-\sqrt{21}/2[/itex] to [itex]\sqrt{21}/2[/itex] and, for each x, y from [itex]-3/2- \sqrt{21/4- x^2}[/itex] to [itex]-3/2+ \sqrt{21/4- x^2}[/itex].